Sure, things will be different in the future, but that doesn't mean that asteroid mining will become profitable. That's just magic thinking. Despite things being a whole lot different than 9,000 years ago, asteroid mining is still just as far away. You're making the invalid assumption that exponential curves can be extrapolated arbitrarily into the future. It's a very common mistake. The recent boom and bust in the housing market is caused by people making the same assumption.
And of course, the desire to have large families doesn't have to be entirely up to genetics. The only requirement is that some genes have some influence. Families with those genes will have a slightly larger exponent in their exponential growth. A slightly larger exponent is sufficient to take over the population, given enough time. Also, people tend to look for partners with similar interest in family size, so different genes contributing to large family sizes will be combined in their children, giving them even more desire to have bigger families.